nLab universal coacting bialgebra

There are two closely related constructions, due Yuri Manin, for finitely generated quadratic algebras, and due Tambara, for finite dimensional algebras.

Tambara’s universal coacting bialgebra

If AA is a finite dimensional (associative unital) kk-algebra, and DD the functor DDAD\mapsto D\otimes A where DD is a kk-algebras has a left adjoint a(A,)a(A,-).

Hom kalg(B,AD)Hom kalg(a(A,B),D) Hom_{k-alg}(B,A\otimes D) \cong Hom_{k-alg}(a(A,B),D)

where A,DA,D are arbitrary kk-algebras. a(A,A)a(A,A) has a canonical structure of a coalgebra, making it into a kk-bialgebra, the universal coacting bialgebra.

  • Daisuke Tambara, The coendomorphism bialgebra of an algebra, J. Fac. Sci. Univ. Tokyo Sect. IA Math, 37, 425-456, 1990 pdf

Tambara’s construction is dual to the universal measuring coalgebra of Sweedler.

Manin’s universal coacting bialgebra

In a similar way to above, one utilizes the adjunction between inner hom and !! functor for quadratic algebras.

  • Yu. I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.

See also quantum linear group.

Generalizations and analogues

  • A. L. Agore, A. S. Gordienko, Joost Vercruysse, V -universal Hopf algebras (co)acting on Ω-algebras Commun. Contemp. Math. 25 (2023), 2150095.
  • A. L. Agore, Functors between representation categories. Universal modules, arXiv:2301.03051
category: algebra

Created on June 12, 2023 at 12:02:53. See the history of this page for a list of all contributions to it.